CAST: Unifying Penrose Tilings via Cyclotomic Fields and Pisot Numbers — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) unify Penrose-like tilings under algebraic integers of cyclotomic fields, with substitution eigenvalues tied to Pisot numbers. MATH: - Vertices supported on ℚ(ζ₂ₙ), the 2n-th cyclotomic field (ζ₂ₙ = eπi/n). - Substitution matrix eigenvalues are algebraic integers, often Pisot numbers (e.g., τ = (1+√5)/2 ≈ 1.618 for Penrose). - Minimal inflation multiplier for Penrose tiling: τ² = τ + 1 = 2.618. - For n=5 (decagonal symmetry): eigenvalues include τ and its Galois conjugate τ' = 1 - τ ≈ -0.618. CONNECTION: - Golden ratio φ = 1.618 appears as dominant eigenvalue; its reciprocal 0.618 and square 2.618 are intrinsic. - 5-fold rotational symmetry (forbidden in periodic crystals) realized via cyclotomic field ℚ(ζ₁₀). - Base-60 not directly present, but cyclotomic fields relate to regular polygon geometry (e.g., 60° rotations for n=6). DEPTH: 9 — Bridges number theory (algebraic integers, Pisot numbers), cry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com